Theorem fprod2dlemstep 11629

Description: Lemma for fprod2d 11630- induction step. (Contributed by Scott Fenton, 30-Jan-2018.)

Hypotheses

Ref	Expression
fprod2d.1	⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)
fprod2d.2	⊢ (𝜑 → 𝐴 ∈ Fin)
fprod2d.3	⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)
fprod2d.4	⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)
fprod2d.5	⊢ (𝜑 → ¬ 𝑦 ∈ 𝑥)
fprod2d.6	⊢ (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
fprod2dlemstep.x	⊢ (𝜑 → 𝑥 ∈ Fin)
fprod2d.7	⊢ (𝜓 ↔ ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷)

Assertion

Ref	Expression
fprod2dlemstep	⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)

Distinct variable groups:   𝐴,𝑗,𝑘   𝐵,𝑘,𝑧   𝑧,𝐶   𝐷,𝑗,𝑘   𝜑,𝑗   𝑥,𝑗   𝑦,𝑗,𝑧   𝜑,𝑘   𝑥,𝑘   𝑦,𝑘,𝑧   𝜑,𝑧   𝑥,𝑧   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦,𝑧,𝑗,𝑘)   𝐴(𝑥,𝑦,𝑧)   𝐵(𝑥,𝑦,𝑗)   𝐶(𝑥,𝑦,𝑗,𝑘)   𝐷(𝑥,𝑦,𝑧)

Proof of Theorem fprod2dlemstep

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
2		fprod2d.7	. . . 4 ⊢ (𝜓 ↔ ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷)
3	1, 2	sylib 122	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷)
4		nfcv 2319	. . . . . 6 ⊢ Ⅎ𝑚∏𝑘 ∈ 𝐵 𝐶
5		nfcsb1v 3090	. . . . . . 7 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐵
6		nfcsb1v 3090	. . . . . . 7 ⊢ Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐶
7	5, 6	nfcprod 11562	. . . . . 6 ⊢ Ⅎ𝑗∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶
8		csbeq1a 3066	. . . . . . 7 ⊢ (𝑗 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑗⦌𝐵)
9		csbeq1a 3066	. . . . . . . 8 ⊢ (𝑗 = 𝑚 → 𝐶 = ⦋𝑚 / 𝑗⦌𝐶)
10	9	adantr 276	. . . . . . 7 ⊢ ((𝑗 = 𝑚 ∧ 𝑘 ∈ 𝐵) → 𝐶 = ⦋𝑚 / 𝑗⦌𝐶)
11	8, 10	prodeq12dv 11576	. . . . . 6 ⊢ (𝑗 = 𝑚 → ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶)
12	4, 7, 11	cbvprodi 11567	. . . . 5 ⊢ ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶 = ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶
13		fprod2d.6	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
14	13	unssbd 3313	. . . . . . . 8 ⊢ (𝜑 → {𝑦} ⊆ 𝐴)
15		vex 2740	. . . . . . . . 9 ⊢ 𝑦 ∈ V
16	15	snss 3727	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 ↔ {𝑦} ⊆ 𝐴)
17	14, 16	sylibr 134	. . . . . . 7 ⊢ (𝜑 → 𝑦 ∈ 𝐴)
18		fprod2d.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin)
19	18	ralrimiva 2550	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ Fin)
20		nfcsb1v 3090	. . . . . . . . . . 11 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐵
21	20	nfel1 2330	. . . . . . . . . 10 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐵 ∈ Fin
22		csbeq1a 3066	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑗⦌𝐵)
23	22	eleq1d 2246	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (𝐵 ∈ Fin ↔ ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin))
24	21, 23	rspc 2835	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 𝐵 ∈ Fin → ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin))
25	17, 19, 24	sylc 62	. . . . . . . 8 ⊢ (𝜑 → ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin)
26		fprod2d.4	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ)
27	26	ralrimivva 2559	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
28		nfcsb1v 3090	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐶
29	28	nfel1 2330	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ
30	20, 29	nfralw 2514	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ
31		csbeq1a 3066	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑗⦌𝐶)
32	31	eleq1d 2246	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑦 → (𝐶 ∈ ℂ ↔ ⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ))
33	22, 32	raleqbidv 2684	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → (∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ ↔ ∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ))
34	30, 33	rspc 2835	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ → ∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ))
35	17, 27, 34	sylc 62	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)
36	35	r19.21bi 2565	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) → ⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)
37	25, 36	fprodcl 11614	. . . . . . 7 ⊢ (𝜑 → ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)
38		csbeq1 3060	. . . . . . . . 9 ⊢ (𝑚 = 𝑦 → ⦋𝑚 / 𝑗⦌𝐵 = ⦋𝑦 / 𝑗⦌𝐵)
39		csbeq1 3060	. . . . . . . . . 10 ⊢ (𝑚 = 𝑦 → ⦋𝑚 / 𝑗⦌𝐶 = ⦋𝑦 / 𝑗⦌𝐶)
40	39	adantr 276	. . . . . . . . 9 ⊢ ((𝑚 = 𝑦 ∧ 𝑘 ∈ ⦋𝑚 / 𝑗⦌𝐵) → ⦋𝑚 / 𝑗⦌𝐶 = ⦋𝑦 / 𝑗⦌𝐶)
41	38, 40	prodeq12dv 11576	. . . . . . . 8 ⊢ (𝑚 = 𝑦 → ∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶)
42	41	prodsn 11600	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ) → ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶)
43	17, 37, 42	syl2anc 411	. . . . . 6 ⊢ (𝜑 → ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶)
44		nfcv 2319	. . . . . . . 8 ⊢ Ⅎ𝑚⦋𝑦 / 𝑗⦌𝐶
45		nfcsb1v 3090	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
46		csbeq1a 3066	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → ⦋𝑦 / 𝑗⦌𝐶 = ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
47	44, 45, 46	cbvprodi 11567	. . . . . . 7 ⊢ ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 = ∏𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
48		csbeq1 3060	. . . . . . . . 9 ⊢ (𝑚 = (2nd ‘𝑧) → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
49		snfig 6813	. . . . . . . . . . 11 ⊢ (𝑦 ∈ V → {𝑦} ∈ Fin)
50	49	elv 2741	. . . . . . . . . 10 ⊢ {𝑦} ∈ Fin
51		xpfi 6928	. . . . . . . . . 10 ⊢ (({𝑦} ∈ Fin ∧ ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin) → ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ∈ Fin)
52	50, 25, 51	sylancr 414	. . . . . . . . 9 ⊢ (𝜑 → ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ∈ Fin)
53		2ndconst 6222	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → (2nd ↾ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)):({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)–1-1-onto→⦋𝑦 / 𝑗⦌𝐵)
54	17, 53	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (2nd ↾ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)):({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)–1-1-onto→⦋𝑦 / 𝑗⦌𝐵)
55		fvres 5539	. . . . . . . . . 10 ⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) → ((2nd ↾ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))‘𝑧) = (2nd ‘𝑧))
56	55	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → ((2nd ↾ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))‘𝑧) = (2nd ‘𝑧))
57	45	nfel1 2330	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ
58	46	eleq1d 2246	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ ↔ ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ))
59	57, 58	rspc 2835	. . . . . . . . . 10 ⊢ (𝑚 ∈ ⦋𝑦 / 𝑗⦌𝐵 → (∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ))
60	35, 59	mpan9 281	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ⦋𝑦 / 𝑗⦌𝐵) → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)
61	48, 52, 54, 56, 60	fprodf1o 11595	. . . . . . . 8 ⊢ (𝜑 → ∏𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
62		elxp 4643	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ↔ ∃𝑚∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)))
63		nfv 1528	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗 𝑧 = ⟨𝑚, 𝑘⟩
64		nfv 1528	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗 𝑚 ∈ {𝑦}
65	20	nfcri 2313	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵
66	64, 65	nfan 1565	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗(𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)
67	63, 66	nfan 1565	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))
68	67	nfex 1637	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))
69		nfv 1528	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))
70		opeq1 3778	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑗 → ⟨𝑚, 𝑘⟩ = ⟨𝑗, 𝑘⟩)
71	70	eqeq2d 2189	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑗 → (𝑧 = ⟨𝑚, 𝑘⟩ ↔ 𝑧 = ⟨𝑗, 𝑘⟩))
72		eleq1w 2238	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑗 → (𝑚 ∈ {𝑦} ↔ 𝑗 ∈ {𝑦}))
73		velsn 3609	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ {𝑦} ↔ 𝑗 = 𝑦)
74	72, 73	bitrdi 196	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑗 → (𝑚 ∈ {𝑦} ↔ 𝑗 = 𝑦))
75	74	anbi1d 465	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) ↔ (𝑗 = 𝑦 ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)))
76	22	eleq2d 2247	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑦 → (𝑘 ∈ 𝐵 ↔ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))
77	76	pm5.32i 454	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵) ↔ (𝑗 = 𝑦 ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))
78	75, 77	bitr4di 198	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) ↔ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)))
79	71, 78	anbi12d 473	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑗 → ((𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) ↔ (𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))))
80	79	exbidv 1825	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑗 → (∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) ↔ ∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))))
81	68, 69, 80	cbvexv1 1752	. . . . . . . . . . . 12 ⊢ (∃𝑚∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) ↔ ∃𝑗∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)))
82	62, 81	bitri 184	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ↔ ∃𝑗∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)))
83		nfv 1528	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝜑
84		nfcv 2319	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(2nd ‘𝑧)
85	84, 28	nfcsbw 3093	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
86	85	nfeq2 2331	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
87		nfv 1528	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝜑
88		nfcsb1v 3090	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
89	88	nfeq2 2331	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶
90		fprod2d.1	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)
91	90	ad2antlr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = 𝐶)
92	31	ad2antrl 490	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐶 = ⦋𝑦 / 𝑗⦌𝐶)
93		fveq2 5515	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd ‘𝑧) = (2nd ‘⟨𝑗, 𝑘⟩))
94		vex 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑗 ∈ V
95		vex 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑘 ∈ V
96	94, 95	op2nd 6147	. . . . . . . . . . . . . . . . . 18 ⊢ (2nd ‘⟨𝑗, 𝑘⟩) = 𝑘
97	93, 96	eqtr2di 2227	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝑘 = (2nd ‘𝑧))
98	97	ad2antlr 489	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝑘 = (2nd ‘𝑧))
99		csbeq1a 3066	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (2nd ‘𝑧) → ⦋𝑦 / 𝑗⦌𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
100	98, 99	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → ⦋𝑦 / 𝑗⦌𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
101	91, 92, 100	3eqtrd 2214	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 = ⟨𝑗, 𝑘⟩) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
102	101	expl 378	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶))
103	87, 89, 102	exlimd 1597	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶))
104	83, 86, 103	exlimd 1597	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑗∃𝑘(𝑧 = ⟨𝑗, 𝑘⟩ ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶))
105	82, 104	biimtrid 152	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶))
106	105	imp 124	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → 𝐷 = ⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
107	106	prodeq2dv 11573	. . . . . . . 8 ⊢ (𝜑 → ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)
108	61, 107	eqtr4d 2213	. . . . . . 7 ⊢ (𝜑 → ∏𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)
109	47, 108	eqtrid 2222	. . . . . 6 ⊢ (𝜑 → ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)
110	43, 109	eqtrd 2210	. . . . 5 ⊢ (𝜑 → ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)
111	12, 110	eqtrid 2222	. . . 4 ⊢ (𝜑 → ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)
112	111	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)
113	3, 112	oveq12d 5892	. 2 ⊢ ((𝜑 ∧ 𝜓) → (∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶) = (∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 · ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷))
114		fprod2d.5	. . . . 5 ⊢ (𝜑 → ¬ 𝑦 ∈ 𝑥)
115		disjsn 3654	. . . . 5 ⊢ ((𝑥 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝑥)
116	114, 115	sylibr 134	. . . 4 ⊢ (𝜑 → (𝑥 ∩ {𝑦}) = ∅)
117		eqidd 2178	. . . 4 ⊢ (𝜑 → (𝑥 ∪ {𝑦}) = (𝑥 ∪ {𝑦}))
118		fprod2dlemstep.x	. . . . 5 ⊢ (𝜑 → 𝑥 ∈ Fin)
119	15	a1i 9	. . . . 5 ⊢ (𝜑 → 𝑦 ∈ V)
120		unsnfi 6917	. . . . 5 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ∈ V ∧ ¬ 𝑦 ∈ 𝑥) → (𝑥 ∪ {𝑦}) ∈ Fin)
121	118, 119, 114, 120	syl3anc 1238	. . . 4 ⊢ (𝜑 → (𝑥 ∪ {𝑦}) ∈ Fin)
122	13	sselda 3155	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝑗 ∈ 𝐴)
123	26	anassrs 400	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
124	18, 123	fprodcl 11614	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∏𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
125	122, 124	syldan 282	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → ∏𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
126	116, 117, 121, 125	fprodsplit 11604	. . 3 ⊢ (𝜑 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = (∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶))
127	126	adantr 276	. 2 ⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = (∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶))
128		eliun 3890	. . . . . . . . . 10 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝑥 𝑧 ∈ ({𝑗} × 𝐵))
129		xp1st 6165	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ {𝑗})
130		elsni 3610	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑧) ∈ {𝑗} → (1st ‘𝑧) = 𝑗)
131	129, 130	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) = 𝑗)
132	131	eleq1d 2246	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → ((1st ‘𝑧) ∈ 𝑥 ↔ 𝑗 ∈ 𝑥))
133	132	biimparc 299	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ 𝑥 ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → (1st ‘𝑧) ∈ 𝑥)
134	133	rexlimiva 2589	. . . . . . . . . 10 ⊢ (∃𝑗 ∈ 𝑥 𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ 𝑥)
135	128, 134	sylbi 121	. . . . . . . . 9 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ 𝑥)
136		xp1st 6165	. . . . . . . . 9 ⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) → (1st ‘𝑧) ∈ {𝑦})
137	135, 136	anim12i 338	. . . . . . . 8 ⊢ ((𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → ((1st ‘𝑧) ∈ 𝑥 ∧ (1st ‘𝑧) ∈ {𝑦}))
138		elin 3318	. . . . . . . 8 ⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) ↔ (𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)))
139		elin 3318	. . . . . . . 8 ⊢ ((1st ‘𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ ((1st ‘𝑧) ∈ 𝑥 ∧ (1st ‘𝑧) ∈ {𝑦}))
140	137, 138, 139	3imtr4i 201	. . . . . . 7 ⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → (1st ‘𝑧) ∈ (𝑥 ∩ {𝑦}))
141	116	eleq2d 2247	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ (1st ‘𝑧) ∈ ∅))
142		noel 3426	. . . . . . . . 9 ⊢ ¬ (1st ‘𝑧) ∈ ∅
143	142	pm2.21i 646	. . . . . . . 8 ⊢ ((1st ‘𝑧) ∈ ∅ → 𝑧 ∈ ∅)
144	141, 143	syl6bi 163	. . . . . . 7 ⊢ (𝜑 → ((1st ‘𝑧) ∈ (𝑥 ∩ {𝑦}) → 𝑧 ∈ ∅))
145	140, 144	syl5 32	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → 𝑧 ∈ ∅))
146	145	ssrdv 3161	. . . . 5 ⊢ (𝜑 → (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) ⊆ ∅)
147		ss0 3463	. . . . 5 ⊢ ((∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) ⊆ ∅ → (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) = ∅)
148	146, 147	syl 14	. . . 4 ⊢ (𝜑 → (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) = ∅)
149		iunxun 3966	. . . . . 6 ⊢ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵))
150		nfcv 2319	. . . . . . . . 9 ⊢ Ⅎ𝑚({𝑗} × 𝐵)
151		nfcv 2319	. . . . . . . . . 10 ⊢ Ⅎ𝑗{𝑚}
152	151, 5	nfxp 4653	. . . . . . . . 9 ⊢ Ⅎ𝑗({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)
153		sneq 3603	. . . . . . . . . 10 ⊢ (𝑗 = 𝑚 → {𝑗} = {𝑚})
154	153, 8	xpeq12d 4651	. . . . . . . . 9 ⊢ (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵))
155	150, 152, 154	cbviun 3923	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = ∪ 𝑚 ∈ {𝑦} ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)
156		sneq 3603	. . . . . . . . . 10 ⊢ (𝑚 = 𝑦 → {𝑚} = {𝑦})
157	156, 38	xpeq12d 4651	. . . . . . . . 9 ⊢ (𝑚 = 𝑦 → ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))
158	15, 157	iunxsn 3963	. . . . . . . 8 ⊢ ∪ 𝑚 ∈ {𝑦} ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)
159	155, 158	eqtri 2198	. . . . . . 7 ⊢ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)
160	159	uneq2i 3286	. . . . . 6 ⊢ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵)) = (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))
161	149, 160	eqtri 2198	. . . . 5 ⊢ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))
162	161	a1i 9	. . . 4 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)))
163		snfig 6813	. . . . . . . 8 ⊢ (𝑗 ∈ V → {𝑗} ∈ Fin)
164	163	elv 2741	. . . . . . 7 ⊢ {𝑗} ∈ Fin
165	122, 18	syldan 282	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝐵 ∈ Fin)
166		xpfi 6928	. . . . . . 7 ⊢ (({𝑗} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑗} × 𝐵) ∈ Fin)
167	164, 165, 166	sylancr 414	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → ({𝑗} × 𝐵) ∈ Fin)
168	167	ralrimiva 2550	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
169		disjsnxp 6237	. . . . . 6 ⊢ Disj 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)
170	169	a1i 9	. . . . 5 ⊢ (𝜑 → Disj 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵))
171		iunfidisj 6944	. . . . 5 ⊢ (((𝑥 ∪ {𝑦}) ∈ Fin ∧ ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin ∧ Disj 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)) → ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
172	121, 168, 170, 171	syl3anc 1238	. . . 4 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin)
173		eliun 3890	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ↔ ∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵))
174		elxp 4643	. . . . . . . 8 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) ↔ ∃𝑚∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵)))
175		simprl 529	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑧 = ⟨𝑚, 𝑘⟩)
176		simprrl 539	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑚 ∈ {𝑗})
177		elsni 3610	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ {𝑗} → 𝑚 = 𝑗)
178	176, 177	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑚 = 𝑗)
179	178	opeq1d 3784	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → ⟨𝑚, 𝑘⟩ = ⟨𝑗, 𝑘⟩)
180	175, 179	eqtrd 2210	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑧 = ⟨𝑗, 𝑘⟩)
181	180, 90	syl 14	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝐷 = 𝐶)
182		simpll 527	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝜑)
183	122	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑗 ∈ 𝐴)
184		simprrr 540	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑘 ∈ 𝐵)
185	182, 183, 184, 26	syl12anc 1236	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝐶 ∈ ℂ)
186	181, 185	eqeltrd 2254	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝐷 ∈ ℂ)
187	186	ex 115	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → ((𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵)) → 𝐷 ∈ ℂ))
188	187	exlimdvv 1897	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → (∃𝑚∃𝑘(𝑧 = ⟨𝑚, 𝑘⟩ ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵)) → 𝐷 ∈ ℂ))
189	174, 188	biimtrid 152	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → (𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
190	189	rexlimdva 2594	. . . . . 6 ⊢ (𝜑 → (∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
191	173, 190	biimtrid 152	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) → 𝐷 ∈ ℂ))
192	191	imp 124	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)) → 𝐷 ∈ ℂ)
193	148, 162, 172, 192	fprodsplit 11604	. . 3 ⊢ (𝜑 → ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 · ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷))
194	193	adantr 276	. 2 ⊢ ((𝜑 ∧ 𝜓) → ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (∏𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 · ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷))
195	113, 127, 194	3eqtr4d 2220	1 ⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  Vcvv 2737  ⦋csb 3057   ∪ cun 3127   ∩ cin 3128   ⊆ wss 3129  ∅c0 3422  {csn 3592  ⟨cop 3595  ∪ ciun 3886  Disj wdisj 3980   × cxp 4624   ↾ cres 4628  –1-1-onto→wf1o 5215  ‘cfv 5216  (class class class)co 5874  1^st c1st 6138  2^nd c2nd 6139  Fincfn 6739  ℂcc 7808   · cmul 7815  ∏cprod 11557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-disj 3981  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-frec 6391  df-1o 6416  df-oadd 6420  df-er 6534  df-en 6740  df-dom 6741  df-fin 6742  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-div 8629  df-inn 8919  df-2 8977  df-3 8978  df-4 8979  df-n0 9176  df-z 9253  df-uz 9528  df-q 9619  df-rp 9653  df-fz 10008  df-fzo 10142  df-seqfrec 10445  df-exp 10519  df-ihash 10755  df-cj 10850  df-re 10851  df-im 10852  df-rsqrt 11006  df-abs 11007  df-clim 11286  df-proddc 11558

This theorem is referenced by:  fprod2d  11630